From a commenter on the web, 21 May 2010:
Tampa Bay: Playing .732 ball in the toughest division in baseball, wiped their feet on NY twice. If they sweep Houston, which seems pretty likely, they will be at .750, which I [the commenter] have never heard of.
At the time of that posting, the Rays were 30-11. Quick calculation: if a team is good enough to be expected to win 100 games, that is, Pr(win) = 100/162 = .617, then there’s a 5% chance that they’ll have won at least 30 of their first 41 games. That’s a calculation based on simple probability theory of independent events, which isn’t quite right here but will get you close and is a good way to train one’s intuition, I think.
Having a .732 record after 41 games is not unheard-of. The Detroit Tigers won 35 of their first 40 games in 1984: that’s .875. (I happen to remember that fast start, having been an Orioles fan at the time.)
Now on to the key ideas
The passage quoted above illustrates three statistical fallacies which I believe are common but are not often discussed:
1. Conditioning on the extrapolation. “If they sweep Houston . . .” The relevant data were that the Rays were .732, not .750.
2. Counting data twice: “Playing .732 . . . wiped their feet on NY twice.” Beating the Yankees is part of how they got to .732 in the first place.
3. Remembered historical evidence: “at .750, which I have never heard of.” There’s no particular reason the commenter should’ve heard of the 1984 Tigers; my point here is that past data aren’t always as you remember them.
P.S. I don’t mean to pick on the above commenter, who I’m sure was just posting some idle thoughts. In some ways, though, perhaps these low-priority remarks are the best windows into our implicit thinking.
P.P.S. Yes, I realize this is out of date–the perils of lagged blog posting. But the general statistical principles are still valid.
add to this list, the unwarranted extrapolation, in sports, they typically say "at this rate" when such rate is known to be highly variable. That's also when the computer tells us remaining download time is 1 hour (based on the instantaneous download speed), then several minutes later, the remaining download time increased to 1:30 when logic says time should go down, and these forecasts are 100% accurate as time -> time of completion.
I don't believe that #1 and #2 are fallacies. Or rather, maybe they are fallacies in general, the original comment doesn't display them.
#1. The relevant data is not just their win-loss record, but also their strength of schedule. Here I assume the commenter is saying that Tampa is about to play a weak team, so their record is likely to improve further.
#2. This isn't counting data twice, it's adding new data. The commenter is saying, "Not only do they have a great record, but they weren't just playing patsies. They had to beat some good teams to get to their current record."
Now, #3 is of course a fair point.
Isn't the main issue here that not all .750s are the same? Presumably the commenter is comparing .750 with a sample size of 182 (a complete season), and .750 with a size of 40. It's natural for the smaller samples to be far more volatile then the larger ones. Remember that after their first games, half the teams will be at 1.0! Unheard-of!